Convolutional Neural Networks (CNNs) are highly effective for image reconstruction problems. Typically, CNNs are trained on large amounts of training images. Recently, however, un-trained CNNs such as the Deep Image Prior and Deep Decoder have achieved excellent performance for image reconstruction problems such as denoising and inpainting, without using any training data. In this work, we study the performance of un-trained neural networks for the reconstruction problem arising in accelerated Magnetic Resonance Imaging (MRI). For this purpose, we view the performance from two perspectives: reconstruction accuracy, and reconstruction reliability. Reconstruction accuracy: One of the main goals in solving image reconstruction tasks is obtaining a high-quality image. In order to measure the quality of the images reconstructed by an un-trained network, we first propose a highly-optimized un-trained recovery approach based on a variation of the Deep Decoder. Afterward, through extensive experiments, we show that the resulting method significantly outperforms conventional un-trained methods such as total-variation norm minimization, as well as naive applications of un-trained networks. Most importantly, the proposed approach achieves on-par performance with a standard trained baseline, the U-net, on the FastMRI dataset, a dataset for benchmarking deep-learning based reconstruction methods. This demonstrates that there is less benefit in learning a prior for solving the accelerated MRI reconstruction problem. This conclusion is drawn by comparison with a baseline trained neural network, however, state-of-the-art methods still slightly outperform U-net (and hence the proposed approach in this work). Reconstruction reliability: Recent works on accelerated MRI reconstruction suggest that trained neural networks are not reliable for image reconstruction tasks, albeit achieving excellent accuracy. For example, at the inference time, small changes (also referred to as adversarial perturbations) in the input of the network can result in significant reconstruction artifacts. In this regard, we analyze the robustness of trained and un-trained methods. Specifically, we consider three notions of robustness: (i) robustness against small changes in the input, (ii) robustness in recovering small details in the image, and (iii) robustness to distribution shifts. Our main findings from this analysis are the followings: (i) contrary to the current belief, neither of the trained and un-trained methods is robust to small changes in the input, and (ii) in opposition to trained neural networks, un-trained methods are naturally robust to data distribution shifts, and interestingly, an un-trained neural network outperforms a trained one after the distribution shift. This work promotes the use of un-trained neural networks for accelerated MRI reconstruction through the following conclusions. First, in terms of accuracy, un-trained neural networks yield high-quality reconstructions, significantly better than conventional un-trained methods and similar to baseline trained methods. Second, a key advantage of un-trained networks over trained ones is a better generalization to unseen data distributions.